K11a74

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K11a73.gif

K11a73

K11a75.gif

K11a75

Contents

K11a74.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a74 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,8,15,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 18 2 6 20 22 8 16
A Braid Representative
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A Morse Link Presentation K11a74 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a74/ThurstonBennequinNumber
Hyperbolic Volume 12.3052
A-Polynomial See Data:K11a74/A-polynomial

[edit Notes for K11a74's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -4

[edit Notes for K11a74's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+10 t^2-13 t+15-13 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 73, 4 }
Jones polynomial -q^9+3 q^8-5 q^7+8 q^6-10 q^5+11 q^4-11 q^3+9 q^2-7 q+5-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-15 z^2 a^{-2} +13 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-7 a^{-2} +5 a^{-4} - a^{-6} +4
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +z^8 a^{-2} +7 z^8 a^{-4} +7 z^8 a^{-6} +z^8-10 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} -23 z^6 a^{-2} -40 z^6 a^{-4} -16 z^6 a^{-6} +7 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +18 z^5 a^{-3} -18 z^5 a^{-5} -16 z^5 a^{-7} +5 z^5 a^{-9} +47 z^4 a^{-2} +51 z^4 a^{-4} +5 z^4 a^{-6} -9 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-6 z^3 a^{-1} +8 z^3 a^{-3} +26 z^3 a^{-5} +8 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -32 z^2 a^{-2} -23 z^2 a^{-4} +2 z^2 a^{-8} -z^2 a^{-10} -12 z^2-z a^{-1} -7 z a^{-3} -9 z a^{-5} -3 z a^{-7} +7 a^{-2} +5 a^{-4} + a^{-6} +4
The A2 invariant q^6+q^4+q^2+2- q^{-2} -2 q^{-6} -2 q^{-8} + q^{-10} -2 q^{-12} +3 q^{-14} + q^{-18} + q^{-20} - q^{-22} + q^{-24} - q^{-26}
The G2 invariant q^{26}-q^{24}+5 q^{22}-7 q^{20}+10 q^{18}-9 q^{16}+2 q^{14}+15 q^{12}-31 q^{10}+46 q^8-42 q^6+23 q^4+15 q^2-53+84 q^{-2} -81 q^{-4} +53 q^{-6} -2 q^{-8} -52 q^{-10} +82 q^{-12} -81 q^{-14} +50 q^{-16} -4 q^{-18} -40 q^{-20} +57 q^{-22} -50 q^{-24} +11 q^{-26} +25 q^{-28} -58 q^{-30} +60 q^{-32} -40 q^{-34} -7 q^{-36} +51 q^{-38} -87 q^{-40} +94 q^{-42} -70 q^{-44} +20 q^{-46} +40 q^{-48} -85 q^{-50} +103 q^{-52} -82 q^{-54} +41 q^{-56} +18 q^{-58} -54 q^{-60} +67 q^{-62} -48 q^{-64} +15 q^{-66} +24 q^{-68} -40 q^{-70} +35 q^{-72} -9 q^{-74} -21 q^{-76} +43 q^{-78} -46 q^{-80} +32 q^{-82} -8 q^{-84} -19 q^{-86} +35 q^{-88} -43 q^{-90} +38 q^{-92} -25 q^{-94} +9 q^{-96} +5 q^{-98} -20 q^{-100} +27 q^{-102} -31 q^{-104} +28 q^{-106} -18 q^{-108} +7 q^{-110} +7 q^{-112} -19 q^{-114} +23 q^{-116} -23 q^{-118} +18 q^{-120} -8 q^{-122} +7 q^{-126} -12 q^{-128} +13 q^{-130} -10 q^{-132} +7 q^{-134} -2 q^{-136} - q^{-138} +2 q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (-2, -1)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
-8 -8 32 \frac{164}{3} \frac{52}{3} 64 \frac{208}{3} -\frac{128}{3} 24 -\frac{256}{3} 32 -\frac{1312}{3} -\frac{416}{3} -\frac{8911}{15} -\frac{5276}{15} -\frac{5044}{45} \frac{127}{9} \frac{209}{15}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=4 is the signature of K11a74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         31 -2
13        52  3
11       53   -2
9      65    1
7     55     0
5    46      -2
3   46       2
1  13        -2
-1 14         3
-3 1          -1
-51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

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K11a73

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K11a75